Conductor (algebraic number theory)

In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.

Let L/K be a finite abelian extension of non-archimedean local fields. The conductor of L/K, denoted ${\displaystyle {\mathfrak {f}}(L/K)}$, is the smallest non-negative integer n such that the higher unit group

is contained in N_L/K(L^×), where N_L/K is field norm map and ${\displaystyle {\mathfrak {m}}_{K}}$ is the maximal ideal of K. Equivalently, n is the smallest integer such that the local Artin map is trivial on ${\displaystyle U_{K}^{(n)}}$. Sometimes, the conductor is defined as ${\displaystyle {\mathfrak {m}}_{K}^{n}}$ where n is as above.

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